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- Open Access

# Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property

- Ravi P Agarwal
^{1, 2}, - Wutiphol Sintunavarat
^{3}Email author and - Poom Kumam
^{3}Email author

**2013**:22

https://doi.org/10.1186/1687-1812-2013-22

© Agarwal et al.; licensee Springer. 2013

**Received:**7 July 2012**Accepted:**15 January 2013**Published:**31 January 2013

## Abstract

In this paper, we prove the coupled coincidence point theorems for a ${w}^{\ast}$-compatible mapping in partially ordered cone metric spaces over a solid cone without the mixed *g*-monotone property. In the case of a totally ordered space, these results are automatically obvious under the assumption given. Therefore, these results can be applied in a much wider class of problems. We also prove the uniqueness of a common coupled fixed point in this setup and give some example which is not applied to the existence of a common coupled fixed point by using the mixed *g*-monotone property but can be applied to our results.

**MSC:**47H10, 54H25.

## Keywords

- cone metric spaces
- common coupled fixed point
- coupled coincidence point
- ${w}^{\ast}$-compatible mappings
- mixed
*g*-monotone property

## 1 Introduction

The famous Banach contraction principle states that if $(X,d)$ is a complete metric space and $T:X\to X$ is a contraction mapping (*i.e.*, $d(Tx,Ty)\le \alpha d(x,y)$ for all $x,y\in X$, where *α* is a non-negative number such that $\alpha <1$), then *T* has a unique fixed point. This principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the various branches of mathematics, but also in economics, chemistry, biology, computer science, engineering, and others. Due to the importance, generalizations of Banach’s contraction principle have been investigated heavily by several authors.

Following this trend, the problem of existence and uniqueness of fixed points in partially ordered sets has been studied thoroughly because of its interesting nature. In 1986, Turinici [1] presented the first result in this direction. Afterward, Ran and Reurings [2] gave some applications of Turinici’s theorem to matrix equations. The results of Ran and Reurings were further extended to ordered cone metric spaces in [3–5]. In 2005, Nieto and Rodríguez-López [6] extended Ran and Reurings’s theorems for nondecreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

*et al.*[9] (see also [10, 11]) studied the existence and uniqueness of solutions of a nonlinear integral equation as an application of coupled fixed points. Very recently, motivated by the work of Caballero

*et al.*[12], Jleli and Samet [13] discussed the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

where $\alpha \in \mathbb{R}$ such that $3<\alpha \le 4$, ${D}_{{0}^{+}}^{\alpha}$ is the Riemann-Liouville fractional derivative and $f:(0,1]\times [0,\mathrm{\infty})\times [0,\mathrm{\infty})\to [0,\mathrm{\infty})$ is continuous, ${lim}_{t\to {0}^{+}}f(t,\cdot ,\cdot )=+\mathrm{\infty}$ (*f* is singular at $t=0$) for all $t\in (0,1]$, $f(t,\cdot ,\cdot )$ is nondecreasing with respect to the first component and decreasing with respect to its second and third components.

Since their important role in the study of the existence and uniqueness of a solution of the periodic boundary value problem, a nonlinear integral equation, and the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem, a wide discussion on coupled fixed point theorems aimed the interest of many scientists.

In 2009, Lakshmikantham and Ćirić [14] extended the concept of a mixed monotone property to a mixed *g*-monotone mapping and proved coupled coincidence point and common coupled fixed point theorems which are more general than the result of Bhaskar and Lakshmikantham in [8]. A number of articles on coupled fixed point, coupled coincidence point, and common coupled fixed point theorems have been dedicated to the improvement; see [15–30] and the references therein.

On the other hand, in 2007, Huang and Zhang [31] have re-introduced the concept of a cone metric space which is replacing the set of real numbers by an ordered Banach space *E*. They went further and defined the convergence via interior points of the cone by which the order in *E* is defined. This approach allows the investigation of cone spaces in the case when the cone is not necessarily normal. They also continued with results concerned with the normal cones only. One of the main results from [31] is the Banach contraction principle in the setting of normal cone spaces. Afterward, many authors generalized their fixed point theorems in cone spaces with normal cones. In other words, the fixed point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is non-normal but just has interior that is nonempty. In this case only, proper generalizations of results from the ordinary metric spaces can be obtained. In 2011, Janković *et al.* [32] gave some examples showing that theorems from ordinary metric spaces cannot be applied in the setting of cone metric spaces, when the cone is non-normal.

Recently, Nashine *et al.* [33] established common coupled fixed point theorems for mixed *g*-monotone and ${w}^{\ast}$-compatible mappings satisfying more general contractive conditions in ordered cone metric spaces over a cone that is only solid (*i.e.*, has a nonempty interior) which improve works of Karapınar [34] and Shatanawi [35]. This result is an ordered version extension of the results of Abbas *et al.* [36].

In this work, we show that the mixed *g*-monotone property in common coupled fixed point theorems in ordered cone metric spaces can be replaced by another property due to Ðorić *et al.* [37]. This property is automatically satisfied in the case of a totally ordered space. Therefore, these results can be applied in a much wider class of problems. Our results generalize and extend many well-known comparable results in the literature. An illustrative example is presented in this work when our results can be used in proving the existence of a common coupled fixed point, while the results of Nashine *et al.* [33] cannot.

## 2 Preliminaries

*E*be a real Banach space with respect to a given norm ${\parallel \cdot \parallel}_{E}$ and let ${0}_{E}$ be a zero vector of

*E*. A nonempty subset

*P*of

*E*is called a

*cone*if the following conditions hold:

- 1.
*P*is closed and $P\ne \{{0}_{E}\}$; - 2.
$a,b\in \mathbb{R}$, $a,b\ge 0$, $x,y\in P\u27f9ax+by\in P$;

- 3.
$x\in P$, $-x\in P\u27f9x={0}_{E}$.

Given a cone $P\subset E$, a partial ordering ${\le}_{P}$ with respect to *P* is naturally defined by $x{\le}_{P}y$ if and only if $y-x\in P$ for $x,y\in E$. We will write $x{<}_{P}y$ to indicate that $x{\le}_{P}y$ but $x\ne y$, while $x\ll y$ will stand for $y-x\in int(P)$, where $int(P)$ denotes the interior of *P*.

*P*is said to be

*normal*if there exists a real number $K>0$ such that for all $x,y\in E$,

The least positive number *K* satisfying the above statement is called a *normal constant* of *P*. In 2008, Rezapour and Hamlbarani [38] showed that there are no normal cones with a normal constant $K<1$.

In what follows, we always suppose that *E* is a real Banach space with cone *P* satisfying $int(P)\ne \mathrm{\varnothing}$ (such cones are called *solid*).

**Definition 2.1** ([31])

*X*be a nonempty set and $d:X\times X\to E$ satisfy

- 1.
${0}_{E}{\le}_{P}d(x,y)$ for all $x,y\in X$ and $d(x,y)={0}_{E}$ if and only if $x=y$;

- 2.
$d(x,y)=d(y,x)$ for all $x,y\in X$;

- 3.
$d(x,y){\le}_{P}d(x,z)+d(z,y)$ for all $x,y,z\in X$.

Then *d* is called a cone metric on *X* and $(X,d)$ is called a cone metric space.

**Definition 2.2** ([31])

*X*, and $x\in X$.

- 1.
If for every $c\in E$ with ${0}_{E}{\ll}_{P}c$, there is $N\in \mathbb{N}$ such that $d({x}_{n},x){\ll}_{P}c$ for all $n\ge N$, then $\{{x}_{n}\}$ is said to

*converge*to*x*. This limit is denoted by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$ or ${x}_{n}\to x$ as $n\to \mathrm{\infty}$. - 2.
If for every $c\in E$ with ${0}_{E}{\ll}_{P}c$, there is $N\in \mathbb{N}$ such that $d({x}_{n},{x}_{m}){\ll}_{P}c$ for all $n,m>N$, then $\{{x}_{n}\}$ is called a

*Cauchy sequence*in*X*. - 3.
If every Cauchy sequence in

*X*is convergent in*X*, then $(X,d)$ is called a*complete*cone metric space.

Let $(X,d)$ be a cone metric space. Then the following properties are often used (particularly when dealing with cone metric spaces in which the cone need not be normal):

(${p}_{1}$) if $a{\le}_{P}ka$, where $a\in P$ and $k\in [0,1)$, then $a={0}_{E}$;

(${p}_{2}$) if ${0}_{E}{\le}_{P}u\ll c$ for each ${0}_{E}\ll c$, then $u={0}_{E}$;

(${p}_{3}$) if $u,v,w\in E$, $u{\le}_{P}v$ and $v\ll w$, then $u\ll w$;

(${p}_{4}$) if $c\in int(P)$, ${0}_{E}{\le}_{P}{a}_{n}\in E$ and ${a}_{n}\to {0}_{E}$, then there exists $k\in \mathbb{N}$ such that for all $n>k$, we have ${a}_{n}\ll c$.

**Definition 2.3**Let

*X*be a nonempty set. Then $(X,d,\u2aaf)$ is called an

*ordered cone metric space*if

- (i)
$(X,d)$ is a cone metric space,

- (ii)
$(X,\u2aaf)$ is a partially ordered set.

Let $(X,\u2aaf)$ be a partially ordered set. By $x\u2ab0y$, we mean $y\u2aafx$ for $x,y\in X$. Elements $x,y\in X$ are called *comparable* if $x\u2aafy$ or $y\u2aafx$ holds. A mapping *f* is said to be *g*-*nondecreasing* (resp., *g*-*nonincreasing*) if, for all $x,y\in X$, $gx\u2aafgy$ implies $f(x)\u2aaff(y)$ (resp., $f(y)\u2aaff(x)$). If *g* is the identity mapping, then *f* is said to be *nondecreasing* (resp., *nonincreasing*).

*F*is said to have a

*mixed*

*g*-

*monotone property*if

*F*is monotone

*g*-nondecreasing in its first argument and monotone

*g*-nonincreasing in its second argument, that is, for any $x,y\in X$,

hold. If in the previous relations *g* is the identity mapping, then it is said that *F* has a *mixed monotone property*.

Let *X* be a nonempty set and $F:X\times X\to X$, $g:X\to X$. An element $(x,y)\in X\times X$ is called

(C_{1}) a *coupled fixed point* of *F* if $x=F(x,y)$ and $y=F(y,x)$;

_{2}) a

*coupled coincidence point*of mappings

*g*and

*F*if

and in this case $(gx,gy)$ is called a *coupled point of coincidence*;

_{3}) a

*common coupled fixed point*of mappings

*g*and

*F*if

**Definition 2.6** ([36])

Let *X* be a nonempty set. Mappings $F:X\times X\to X$ and $g:X\to X$ are called

(${W}_{1}$) *w*-*compatible* if $gF(x,y)=F(gx,gy)$ whenever $gx=F(x,y)$ and $gy=F(y,x)$;

(${W}_{2}$) ${w}^{\ast}$-*compatible* if $gF(x,x)=F(gx,gx)$ whenever $gx=F(x,x)$.

It is easy to see that *w*-compatible implies ${w}^{\ast}$-compatible. The following example shows that the converse of the above argument is not true.

**Example 2.7**Let $X=[0,\mathrm{\infty})$ and $F:X\times X\to X$ and $g:X\to X$ be defined by

It is easy to see that $g0=\pi =F(0,1)$ and $g1=2=F(1,0)$, but $gF(0,1)=5\ne 8=F(g0,g1)$. Hence, *F* and *g* are not *w*-compatible.

However, $F(x,x)=gx$ is possible only if $x\in \{4,6,8,\dots \}$ and for all points in this case, we get $gF(x,x)=8=F(gx,gx)$. Therefore, *F* and *g* are ${w}^{\ast}$-compatible.

For elements *x*, *y* of a partially ordered set $(X,\u2aaf)$, we will write $x\asymp y$ whenever *x* and *y* are comparable (*i.e.*, $x\u2aafy$ or $y\u2aafx$ holds).

Next, we give a new property due to Ðorić *et al.* [37].

*X*be a nonempty set and let $g:X\to X$ and $F:X\times X\to X$. We will consider the following condition:

**Remark 2.8** We obtain that the conditions (2.3) and (2.4) are trivially satisfied if $(X,\u2aaf)$ is the totally ordered.

The following examples show that the condition (2.3) ((2.4), resp.) may be satisfied when *F* does not have the mixed *g*-monotone property (monotone property, resp.).

**Example 2.9**Let $X=\{a,b,c,d\}$, $\u2aaf=\{(a,a),(b,b),(c,c),(d,d),(a,b),(c,d)\}$,

*F*does not have the mixed

*g*-monotone property. But it has property (2.3) since

- (1)
For each $y\in X$, we get $gc\asymp F(c,y)=gc$ and $F(c,y)\asymp F(c,v)$ for all $v\in X$.

- (2)
For each $y\in X$, we get $gd\asymp F(d,y)=gd$ and $F(d,y)\asymp F(d,v)$ for all $v\in X$.

**Example 2.10**Let $X=\{a,b,c,d\}$, $\u2aaf=\{(a,a),(b,b),(c,c),(d,d),(a,b),(c,d)\}$,

*F*does not have the mixed monotone property. But it has property (2.4) since

- (1)
For each $y\in X$, we get $a\asymp F(a,y)$ and $F(a,y)=b\asymp a=F(F(a,y),v)$ for all $v\in X$.

- (2)
For each $y\in X$, we get $b\asymp F(b,y)$ and $F(b,y)=a\asymp b=F(F(b,y),v)$ for all $v\in X$.

- (3)
The other two cases are trivial.

## 3 Coupled coincidence point theorems lacking the mixed *g*-monotone property

In this section, we give the existence of coupled coincidence point theorems in ordered cone metric spaces lacking the mixed *g*-monotone property. Our first main result is the following theorem.

**Theorem 3.1**

*Let*$(X,d,\u2aaf)$

*be an ordered cone metric space over a solid cone*

*P*

*and let*$g:X\to X$

*and*$F:X\times X\to X$.

*Suppose that the following hold*:

- (i)
$F(X\times X)\subseteq g(X)$

*and*$g(X)$*is a complete subspace of**X*; - (ii)
*g**and**F**satisfy property*(2.3); - (iii)
*there exist*${x}_{0},{y}_{0}\in X$*such that*$g{x}_{0}\asymp F({x}_{0},{y}_{0})$*and*$g{y}_{0}\asymp F({y}_{0},{x}_{0})$; - (iv)

*holds*;

- (v)
*if*${x}_{n}\to x$*when*$n\to \mathrm{\infty}$*in**X*,*then*${x}_{n}\asymp x$*for**n**sufficiently large*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *and* *g* *have a coupled coincidence point* $(x,y)\in X\times X$.

*Proof*Starting from ${x}_{0}$, ${y}_{0}$ (condition (iii)) and using the fact that $F(X\times X)\subseteq g(X)$ (condition (i)), we can construct sequences $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ in

*X*such that

since ${\sum}_{i=1}^{6}{a}_{i}<1$.

If $d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})={0}_{E}$, then $({x}_{0},{y}_{0})$ is a coupled coincidence point of *F* and *g*. So, let ${0}_{E}{<}_{P}d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})$.

Since $\frac{{\lambda}^{n}}{1-\lambda}\to 0$ as $n\to \mathrm{\infty}$, we get $\frac{{\lambda}^{n}}{1-\lambda}[d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})]\to {0}_{E}$ as $n\to \mathrm{\infty}$.

*n*,

then by (${p}_{3}$), we get $d(g{x}_{n},g{x}_{m})\ll c$ and $d(g{y}_{n},g{y}_{m})\ll c$ for *n* large enough. Therefore, we get $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ are Cauchy sequences in $g(X)$. By completeness of $g(X)$, there exist $gx,gy\in g(X)$ such that $g{x}_{n}\to gx$ and $g{y}_{n}\to gy$ as $n\to \mathrm{\infty}$.

By (v), we have $g{x}_{n}\asymp gx$ and $gy\asymp g{y}_{n}$ for all $n\ge 0$. Now, we prove that $F(x,y)=gx$ and $F(y,x)=gy$.

Now, according to (${p}_{2}$), it follows that $d(F(x,y),gx)={0}_{E}$ and $F(x,y)=gx$. Similarly, we can prove that $F(y,x)=gy$. Hence, $(x,y)$ is a coupled coincidence point of the mappings *F* and *g*.

Now, according to (${p}_{2}$), it follows that $d(F(x,y),gx)={0}_{E}$ and $F(x,y)=gx$. Similarly, $F(y,x)=gy$. Hence, $(x,y)$ is a coupled coincidence point of the mappings *F* and *g*. □

**Remark 3.2** In Theorem 3.1, the condition (ii) is a substitution for the mixed *g*-monotone property that has been used in most of the coupled coincidence point theorems so far. Therefore, Theorem 3.1 improves the results of Nashine *et al.* [33]. Moreover, it is an ordered version extension of the results of Abbas *et al.* [36].

**Corollary 3.3**

*Let*$(X,d,\u2aaf)$

*be an ordered cone metric space over a solid cone*

*P*

*and let*$g:X\to X$

*and*$F:X\times X\to X$.

*Suppose that the following hold*:

- (i)
$F(X\times X)\subseteq g(X)$

*and*$g(X)$*is a complete subspace of**X*; - (ii)
*g**and**F**satisfy property*(2.3); - (iii)
*there exist*${x}_{0},{y}_{0}\in X$*such that*$g{x}_{0}\asymp F({x}_{0},{y}_{0})$*and*$g{y}_{0}\asymp F({y}_{0},{x}_{0})$; - (iv)
*there exist*$\alpha ,\beta ,\gamma \ge 0$*and*$\alpha +\beta +\gamma <1$*such that for all*$x,y,u,v\in X$*satisfying*$gx\asymp gu$*and*$gy\asymp gv$,$d(F(x,y),F(u,v)){\le}_{P}\alpha d(gx,gu)+\beta d(gy,gv)+\gamma d(F(x,y),gu)$(3.9)

*holds*;

- (v)
*if*${x}_{n}\to x$*when*$n\to \mathrm{\infty}$*in**X*,*then*${x}_{n}\asymp x$*for**n**sufficiently large*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *and* *g* *have a coupled coincidence point* $(x,y)\in X\times X$.

Putting $g={I}_{X}$, where ${I}_{X}$ is the identity mapping from *X* into *X* in Theorem 3.1, we get the following corollary.

**Corollary 3.4**

*Let*$(X,d,\u2aaf)$

*be an ordered cone metric space over a solid cone*

*P*

*and let*$F:X\times X\to X$.

*Suppose that the following hold*:

- (i)
*X**is complete*; - (ii)
*g**and**F**satisfy property*(2.4); - (iii)
*there exist*${x}_{0},{y}_{0}\in X$*such that*${x}_{0}\asymp F({x}_{0},{y}_{0})$*and*${y}_{0}\asymp F({y}_{0},{x}_{0})$; - (iv)

*holds*;

- (v)
*if*${x}_{n}\to x$*when*$n\to \mathrm{\infty}$*in**X*,*then*${x}_{n}\asymp x$*for**n**sufficiently large*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *has a coupled fixed point* $(x,y)\in X\times X$.

Our second main result is the following.

**Theorem 3.5**

*Let*$(X,d,\u2aaf)$

*be an ordered cone metric space over a solid cone*

*P*.

*Let*$F:X\times X\to X$

*and*$g:X\to X$

*be mappings*.

*Suppose that the following hold*:

- (i)
$F(X\times X)\subseteq g(X)$

*and*$g(X)$*is a complete subspace of**X*; - (ii)
*g**and**F**satisfy property*(2.3); - (iii)
*there exist*${x}_{0},{y}_{0}\in X$*such that*$g{x}_{0}\asymp F({x}_{0},{y}_{0})$*and*$g{y}_{0}\asymp F({y}_{0},{x}_{0})$; - (iv)
*there is some*$h\in [0,1/2)$*such that for all*$x,y,u,v\in X$*satisfying*$gx\asymp gu$*and*$gy\asymp gv$,*there exists*${\mathrm{\Theta}}_{x,y,u,v}\in \{d(gx,gu),d(gy,gv),d(F(x,y),gu)\}$

*such that*

- (v)
*if*${x}_{n}\to x$*when*$n\to \mathrm{\infty}$*in**X*,*then*${x}_{n}\asymp x$*for**n**sufficiently large*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *and* *g* *have a coupled coincidence point* $(x,y)\in X\times X$.

*Proof*Since $F(X\times X)\subseteq g(X)$ (condition (i)), we can start from ${x}_{0}$, ${y}_{0}$ (condition (iii)) and construct sequences $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ in

*X*such that

By repeating this process, we have $g{x}_{n-1}\asymp g{x}_{n}$. Similarly, we can prove that $g{y}_{n-1}\asymp g{y}_{n}$ for all $n\in \mathbb{N}$.

Now, denote ${\delta}_{n}=d(g{x}_{n},g{x}_{n+1})+d(g{y}_{n},g{y}_{n+1})$. Since the cases ${\mathrm{\Theta}}_{1}={0}_{E}$ and ${\mathrm{\Theta}}_{2}={0}_{E}$ are trivial, we have to consider the following four possibilities.

Case 3. $d(g{x}_{n},g{x}_{n+1}){\le}_{P}hd(g{y}_{n-1},g{y}_{n})$ and $d(g{y}_{n},g{y}_{n+1}){\le}_{P}hd(g{x}_{n-1},g{x}_{n})$. This case is treated analogously to Case 1.

Case 4. $d(g{x}_{n},g{x}_{n+1}){\le}_{P}hd(g{y}_{n-1},g{y}_{n})$ and $d(g{y}_{n},g{y}_{n+1}){\le}_{P}hd(g{y}_{n-1},g{y}_{n})$. This case is treated analogously to Case 2.

and by the same argument as in Theorem 3.1, it is proved that $\{g{x}_{n}\}$ and $\{g{y}_{n}\}$ are Cauchy sequences in $g(X)$. By the completeness of $g(X)$, there exist $gx,gy\in g(X)$ such that $g{x}_{n}\to gx$ and $g{y}_{n}\to gy$.

From (v), we get $g{x}_{n}\asymp gx$ and $gy\asymp g{y}_{n}$ for all $n\ge 0$. Now, we prove that $F(x,y)=gx$ and $F(y,x)=gy$.

*n*sufficiently large, we have that

*n*sufficiently large,

Therefore, again by property (${p}_{2}$), we get that $F(x,y)=gx$. Similarly, we can prove that $F(y,x)=gy$. Hence, $(x,y)$ is a coupled point of coincidence of *F* and *g*.

*n*sufficiently large, we have that

*n*sufficiently large,

Thus, again by property (${p}_{2}$), we get that $F(x,y)=gx$.

Similarly, $F(y,x)=gy$ is obtained. Hence, $(x,y)$ is a coupled point of coincidence of the mappings *F* and *g*. □

**Remark 3.6** It would be interesting to relate our Theorem 3.5 with Theorem 2.1 of Long *et al.* [39].

Putting $g={I}_{X}$, where ${I}_{X}$ is the identity mapping from *X* into *X* in Theorem 3.5, we get the following corollary.

**Corollary 3.7**

*Let*$(X,d,\u2aaf)$

*be an ordered cone metric space over a solid cone*

*P*.

*Let*$F:X\times X\to X$

*be mappings*.

*Suppose that the following hold*:

- (i)
*X**is complete*; - (ii)
*F**satisfies property*(2.4); - (iii)
*there exist*${x}_{0},{y}_{0}\in X$*such that*${x}_{0}\asymp F({x}_{0},{y}_{0})$*and*${y}_{0}\asymp F({y}_{0},{x}_{0})$; - (iv)
*there is some*$h\in [0,1/2)$*such that for all*$x,y,u,v\in X$*satisfying*$x\asymp u$*and*$y\asymp v$,*there exists*${\mathrm{\Theta}}_{x,y,u,v}\in \{d(x,u),d(y,v),d(F(x,y),u)\}$

*such that*

- (v)
*if*${x}_{n}\to x$*when*$n\to \mathrm{\infty}$*in**X*,*then*${x}_{n}\asymp x$*for**n**sufficiently large*.

*Then there exist*$x,y\in X$

*such that*

*that is*, *F* *has a coupled fixed point* $(x,y)\in X\times X$.

## 4 Common coupled fixed point theorems lacking the mixed monotone property

Some questions arise naturally from Theorems 3.1 and 3.5. For example, one may ask if there are necessary conditions for the existence and uniqueness of a common coupled fixed point of *F* and *g*?

The next theorem provides a positive answer to this question with additional hypotheses to Theorems 3.1 and 3.5.

*X*, we will denote also by ⪯ the order on $X\times X$ given by

**Theorem 4.1**

*In addition to the hypotheses of Theorem*3.1,

*suppose that for every*$(x,y),({x}^{\ast},{y}^{\ast})\in X\times X$,

*there exists*$(u,v)\in X\times X$

*such that*

*and*

*If*

*F*

*and*

*g*

*are*${w}^{\ast}$-

*compatible*,

*then*

*F*

*and*

*g*

*have a unique common coupled fixed point*,

*that is*,

*there exists a unique*$(\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})\in X\times X$

*such that*

*Proof*From Theorem 3.1, the set of coupled coincidence points of

*F*and

*g*is nonempty. Suppose $(x,y)$ and $({x}^{\ast},{y}^{\ast})$ are coupled coincidence points of

*F*, that is, $gx=F(x,y)$, $gy=F(y,x)$, $g{x}^{\ast}=F({x}^{\ast},{y}^{\ast})$ and $g{y}^{\ast}=F({y}^{\ast},{x}^{\ast})$. We will prove that

*n*. Further, set ${x}_{0}=x$, ${y}_{0}=y$, ${x}_{0}^{\ast}={x}^{\ast}$, ${y}_{0}^{\ast}={y}^{\ast}$ and, in a similar way, define the sequences $\{g{x}_{n}\}$, $\{g{y}_{n}\}$ and $\{g{x}_{n}^{\ast}\}$, $\{g{y}_{n}^{\ast}\}$. Then it is easy to show that

as $n\to \mathrm{\infty}$.

where *λ* is determined as in (3.8), and hence $0\le \lambda <1$.

*n*. Since

it follows by (${p}_{3}$) that $d(g{u}_{n},gx)\ll c$ for large *n*, and so $g{u}_{n}\to gx$ when $n\to \mathrm{\infty}$. Similarly, $g{v}_{n}\to gy$ when $n\to \mathrm{\infty}$. By the same procedure, one can show that $g{u}_{n}\to g{x}^{\ast}$ and $g{v}_{n}\to g{y}^{\ast}$ as $n\to \mathrm{\infty}$. By the uniqueness of the limit, we get $gx=g{x}^{\ast}$ and $gy=g{y}^{\ast}$, *i.e.*, (4.2) is proved. Therefore, $(gx,gy)$ is the unique coupled point of coincidence of *F* and *g*.

Note that if $(gx,gy)$ is a coupled point of coincidence of *F* and *g*, then $(gy,gx)$ is also a coupled point of coincidence of *F* and *g*. Then $gx=gy$ and therefore $(gx,gx)$ is the unique coupled point of coincidence of *F* and *g*.

*F*and

*g*have a common coupled fixed point. Let $\stackrel{\u02c6}{u}:=gx$. Then we have $\stackrel{\u02c6}{u}=gx=F(x,x)$. Since

*F*and

*g*are ${w}^{\ast}$-compatible, we have

Thus, $(g\stackrel{\u02c6}{u},g\stackrel{\u02c6}{u})$ is a coupled point of coincidence of *F* and *g*. By the uniqueness of a coupled point of coincidence of *F* and *g*, we get $g\stackrel{\u02c6}{u}=gx$. Therefore, $\stackrel{\u02c6}{u}=g\stackrel{\u02c6}{u}=F(\stackrel{\u02c6}{u},\stackrel{\u02c6}{u})$, that is, $(\stackrel{\u02c6}{u},\stackrel{\u02c6}{u})$ is a common coupled fixed point of *F* and *g*.

*F*and

*g*. Let $(\tilde{u},\tilde{u})\in X\times X$ be another common coupled fixed point of

*F*and

*g*. So,

Then $(g\stackrel{\u02c6}{u},g\stackrel{\u02c6}{u})$ and $(g\tilde{u},g\tilde{u})$ are two common coupled points of coincidence of *F* and *g* and, as was previously proved, it must be $g\stackrel{\u02c6}{u}=g\tilde{u}$, and so $\stackrel{\u02c6}{u}=g\stackrel{\u02c6}{u}=g\tilde{u}=\tilde{u}$. This completes the proof. □

Next, we give some illustrative example which supports Theorem 4.1, while the results of Nashine *et al.* [33] do not.

**Example 4.2**Let $X=\mathbb{R}$ be ordered by the following relation:

for all $x,y\in X$, for a fixed $\phi \in P$ (*e.g.*, $\phi (t)={e}^{t}$ for $t\in [0,1]$). Then $(X,d)$ is a complete ordered cone metric space over a non-normal solid cone.

So, the mapping *F* does not satisfy the mixed *g*-monotone property. Therefore, Theorems 3.1 and 3.2 of Nashine *et al.* [33] cannot be used to reach this conclusion.

Now, we show that Theorem 4.1 can be used for this case.

Take ${a}_{1}={a}_{3}=\frac{1}{4}$ and ${a}_{2}={a}_{4}={a}_{5}={a}_{6}=0$. We will check that the condition (3.1) in Theorem 3.1 holds.

*F*and

*g*are ${w}^{\ast}$-compatible. We note that if $gx=F(x,x)$, then we get only one case, that is, $x=0$, and hence

Therefore, *F* and *g* are ${w}^{\ast}$-compatible.

Moreover, other conditions in Theorem 4.1 are also satisfied. Now, we can apply Theorem 4.1 to conclude the existence of a unique common coupled fixed point of *F* and *g* that is a point $(0,0)$.

The following uniqueness result corresponding to Theorem 3.5 can be proved in the same way as Theorem 4.1.

**Theorem 4.3**

*In addition to the hypotheses of Theorem*3.5,

*suppose that for every*$(x,y),({x}^{\ast},{y}^{\ast})\in X\times X$,

*there exists*$(u,v)\in X\times X$

*such that*

*and*

*If*

*F*

*and*

*g*

*are*${w}^{\ast}$-

*compatible*,

*then*

*F*

*and*

*g*

*have a unique coupled common fixed point*,

*that is*,

*there exists a unique*$(\stackrel{\u02c6}{u},\stackrel{\u02c6}{v})\in X\times X$

*such that*

## Declarations

### Acknowledgements

The authors would like to thank Professor Hichem Ben-El-Mechaiekh and the referee for valuable comments. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the third author would like to thank the Commission on Higher Education, the Thailand Research Fund and KMUTT under Grant No. MRG5580213 for financial support during the preparation of this manuscript.

## Authors’ Affiliations

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